tailieunhanh - Chaotic Systems Part 7

Tham khảo tài liệu 'chaotic systems part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Control and Identification of Chaotic Systems by Altering the Oscillation Energy 139 Another strategy is based on a goal-oriented control of desired target. It can be applied in cases when the system equations are known or a desired target can be identified say extracted from the system time series . The amplitude of system s natural response is derived from equation T 0 x x x F t xcdt . 9 Equation 9 describes the balance of dissipation and supply of system s intrinsic energy. For free self-sustained oscillations this balance is supported entirely by nonlinear damping. To eliminate the natural response distortion imposed by the control the following condition must be satisfied 1 T T yo g x x xdt . 1 For small oscillations substitution of g x x k tanh fix tt k fix 3 fi3x3 into 1 yields fi zb . 11 pw Thus the distortion can be minimized solely by tuning a perturbation shape. If p 1 fi should be sufficiently large so as to preserve the underlying natural response. Control 6-7 does not depend on the type of functions x x x Ị x and F t and hence can be applied to linear and nonlinear oscillators to regular and chaotic dynamics. The approach can be easily generalized to a case of coupled oscillator networks Tereshko et al. 20 4b . 3. Controlling 2-D oscillators Van der Pol oscillator Consider the van der Pol oscillator with x x x x2 ụ x and Ị x x controlled by feedback g x ktanh fix . Linearizing the dynamic equation 1 in the vicinity of x one obtains the eigenvalues A1 ụ kfi ự ụ kfi 2 4 2 and A2 ụ kfi ỵ ụ kfi 2 4 2. At ụ and k the perturbation with k 1 fi ụ or fi 1 k ụ stabilizes the unstable equilibrium. Thus two control strategies can be applied i altering the perturbation magnitude ii reshaping the perturbation. When fi TO tanh fix sign x . The energy change caused by the control yields g x x ksign x x k x . This strategy corresponds to maximizing the rejection injection of the oscillation energy and is in fact the first approximation of optimal control for a van